Subgroups — #27185900

Order 12928.

1424 conjugacy classes of subgroups.

The subgroup lattice has 330 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#20266012	C₂	1	2	yes	maximal, complemented
2	#20265994	C₂	1	1	yes	maximal, characteristic, complemented
3	#20265991	C₂	1	3	yes	maximal, complemented
4	#20265988	C₂	1	2	yes	maximal, complemented
5	#20265987	C₂	1	6	yes	maximal, complemented
6	#20265978	C₂	1	3	yes	maximal, complemented
7	#20265973	C₂	1	6	yes	maximal, complemented
8	C₂×#15692274	C₂	1	3	yes	maximal, complemented
9	#135×D₁₀₁	C₂	1	3	yes	maximal, complemented
10	#131×D₁₀₁	C₂	1	1	yes	maximal, characteristic, complemented
11	#576×C₁₀₁	C₂	1	1	yes	maximal, characteristic, complemented
12	#15692302	—	2	2	no	—
13	#15692302	C₂²	1	2	yes	complemented
14	#15692302	C₂²	1	6	yes	complemented
15	#15692300	—	2	6	no	—
16	#15692300	C₂²	1	6	yes	complemented
17	C₂×#13108073	—	2	6	no	—
18	C₂×#13108073	C₂²	1	6	yes	complemented
19	#41×D₁₀₁	—	2	2	no	—
20	#41×D₁₀₁	C₂²	1	1	yes	characteristic
21	#41×D₁₀₁	C₂²	1	6	yes	complemented
22	#15692288	C₂²	1	6	yes	complemented
23	#15692285	C₂²	1	6	yes	complemented
24	#15692284	C₂²	1	12	yes	complemented
25	#15692281	C₂²	1	6	yes	complemented
26	#15692279	C₂²	1	12	yes	complemented
27	#15692278	C₂²	1	1	yes	characteristic, complemented
28	#15692277	C₂²	1	3	yes	complemented
29	#15692276	C₂²	1	6	yes	complemented
30	#15692275	C₂²	1	6	yes	complemented
31	#15692274	C₂²	1	6	yes	complemented
32	#15692274	C₂²	1	6	yes	complemented
33	#15692274	C₂²	1	3	yes	complemented
34	#15692274	C₂²	1	6	yes	complemented
35	C₂×Dic₄₀₄	C₂²	1	3	yes	complemented
36	Q₁₆×D₁₀₁	C₂²	1	3	yes	complemented
37	C₂×#13108062	C₂²	1	6	yes	complemented
38	SD₁₆×D₁₀₁	C₂²	1	6	yes	complemented
39	D₈×D₁₀₁	C₂²	1	3	yes	complemented
40	#15692263	C₂²	1	3	yes	complemented
41	#15692262	C₂²	1	1	yes	characteristic, complemented
42	#15692261	C₂²	1	3	yes	complemented
43	M₁₆×D₁₀₁	C₂²	1	3	yes	complemented
44	C₂×C₈×D₁₀₁	C₂²	1	3	yes	complemented
45	H(5,𝔽₂)×C₁₀₁	C₂²	1	2	yes	complemented
46	AGL(1,ℤ/8)×C₁₀₁	C₂²	1	6	yes	complemented
47	#135×C₁₀₁	C₂²	1	3	yes	complemented
48	C₂×D₈×C₁₀₁	C₂²	1	3	yes	complemented
49	#131×C₁₀₁	C₂²	1	1	yes	characteristic, complemented
50	#13108075	D₄	1	1	yes	characteristic
51	#13108075	C₂³	1	2	yes	—
52	#13108074	—	2	6	no	—
53	#13108074	—	2	6	no	—
54	#13108074	C₂³	1	3	yes	—
55	#13108074	C₂³	1	6	yes	—
56	#13108073	—	2	6	no	—
57	#13108073	—	2	12	no	—
58	#13108073	—	2	6	no	—
59	#13108073	—	2	6	no	—
60	#13108073	—	2	6	no	—
61	#13108073	—	2	6	no	—
62	#13108073	D₄	1	3	yes	—
63	#13108073	C₂³	1	6	yes	—
64	#13108073	C₂³	1	12	yes	—
65	C₂×Dic₂₀₂	—	2	6	no	—
66	C₂×Dic₂₀₂	C₂³	1	6	yes	—
67	Q₈×D₁₀₁	—	2	2	no	—
68	Q₈×D₁₀₁	D₄	1	1	yes	characteristic
69	Q₈×D₁₀₁	C₂³	1	6	yes	—
70	C₂×C₂²⋊D₁₀₁	—	2	6	no	—
71	C₂×C₂²⋊D₁₀₁	—	2	12	no	—
72	D₄×D₁₀₁	—	2	6	no	—
73	D₄×D₁₀₁	—	2	6	no	—
74	D₄×D₁₀₁	D₄	1	3	yes	—
75	D₄×D₁₀₁	C₂³	1	6	yes	—
76	C₂²×Dic₁₀₁	—	2	6	no	—
77	C₂×C₄×D₁₀₁	—	2	6	no	—
78	C₂×C₄×D₁₀₁	C₂³	1	3	yes	—
79	Dic₄₀₄	C₂³	1	6	yes	—
80	Dic₄₀₄	C₂³	1	3	yes	—
81	#13108064	C₂³	1	6	yes	—
82	#13108063	C₂³	1	6	yes	—
83	#13108062	C₂³	1	6	yes	—
84	#13108062	C₂³	1	12	yes	—
85	#13108061	C₂³	1	2	yes	—
86	D₈₀₈	C₂³	1	1	yes	characteristic
87	#13108059	C₂³	1	6	yes	—
88	#13108058	C₂³	1	3	yes	—
89	#13108057	C₂³	1	6	yes	—
90	C₂×#11717579	C₂³	1	3	yes	—
91	C₈×D₁₀₁	C₂³	1	1	yes	characteristic
92	C₈×D₁₀₁	C₂³	1	3	yes	—
93	C₈×D₁₀₁	C₂³	1	3	yes	—
94	C₈×D₁₀₁	C₂³	1	3	yes	—
95	#41×C₁₀₁	—	2	2	no	—
96	#41×C₁₀₁	C₂³	1	6	yes	—
97	#41×C₁₀₁	C₂³	1	1	yes	characteristic
98	C₂×D₄×C₁₀₁	—	2	6	no	—
99	C₂×D₄×C₁₀₁	C₂³	1	6	yes	—
100	Q₁₆×C₁₀₁	C₂³	1	1	yes	characteristic
101	SD₁₆×C₁₀₁	C₂³	1	6	yes	—
102	D₈×C₁₀₁	C₂³	1	3	yes	—
103	D₈×C₁₀₁	C₂³	1	6	yes	—
104	M₁₆×C₁₀₁	C₂³	1	3	yes	—
105	C₂×C₈×C₁₀₁	C₂³	1	3	yes	—
106	C₂³×C₁₀₁	—	2	6	no	—
107	C₂²×D₁₀₁	—	2	6	no	—
108	C₂²×D₁₀₁	C₂×D₄	1	3	yes	—
109	Q₈×C₁₀₁	C₂×D₄	1	1	yes	characteristic
110	Q₈×C₁₀₁	C₂⁴	1	2	yes	—
111	D₄×C₁₀₁	—	2	6	no	—
112	D₄×C₁₀₁	—	2	6	no	—
113	D₄×C₁₀₁	C₂×D₄	1	3	yes	—
114	D₄×C₁₀₁	C₂⁴	1	6	yes	—
115	C₂×C₄×C₁₀₁	—	2	6	no	—
116	C₂×C₄×C₁₀₁	C₂⁴	1	3	yes	—
117	C₂²⋊D₁₀₁	—	2	6	no	—
118	C₂²⋊D₁₀₁	—	2	12	no	—
119	C₂²⋊D₁₀₁	—	2	6	no	—
120	C₂²⋊D₁₀₁	—	2	6	no	—
121	C₂²⋊D₁₀₁	C₂×D₄	1	6	yes	—
122	C₂×Dic₁₀₁	—	2	6	no	—
123	C₂×Dic₁₀₁	—	2	6	no	—
124	C₂×Dic₁₀₁	—	2	12	no	—
125	C₂×Dic₁₀₁	C₂×D₄	1	3	yes	—
126	D₄₀₄	—	2	2	no	—
127	D₄₀₄	C₂×D₄	1	3	yes	—
128	D₄₀₄	C₂⁴	1	2	yes	—
129	C₄×D₁₀₁	—	2	6	no	—
130	C₄×D₁₀₁	—	2	6	no	—
131	C₄×D₁₀₁	—	2	2	no	—
132	C₄×D₁₀₁	C₂×D₄	1	3	yes	—
133	C₄×D₁₀₁	C₂×D₄	1	3	yes	—
134	C₄×D₁₀₁	C₂⁴	1	1	yes	characteristic
135	C₄×D₁₀₁	C₂⁴	1	3	yes	—
136	Dic₂₀₂	—	2	6	no	—
137	Dic₂₀₂	—	2	6	no	—
138	Dic₂₀₂	C₂×D₄	1	3	yes	—
139	Dic₂₀₂	C₂⁴	1	6	yes	—
140	C₈×C₁₀₁	C₂⁴	1	1	yes	characteristic
141	C₈×C₁₀₁	C₂⁴	1	3	yes	—
142	#11717579	C₂⁴	1	1	yes	characteristic
143	#11717579	C₂⁴	1	3	yes	—
144	C₂²×C₁₀₁	—	2	6	no	—
145	C₂²×C₁₀₁	—	8	6	no	—
146	C₂²×C₁₀₁	C₂²×D₄	1	3	yes	—
147	C₂×D₁₀₁	—	2	2	no	—
148	C₂×D₁₀₁	—	4	3	no	—
149	C₂×D₁₀₁	—	8	6	no	—
150	C₂×D₁₀₁	C₂²×D₄	1	3	yes	—
151	C₂×D₁₀₁	C₂²×D₄	1	1	yes	characteristic
152	C₄×C₁₀₁	—	2	2	no	—
153	C₄×C₁₀₁	C₂²×D₄	1	3	yes	—
154	C₄×C₁₀₁	C₂⁵	1	1	yes	characteristic
155	Dic₁₀₁	—	2	6	no	—
156	Dic₁₀₁	C₂²×D₄	1	3	yes	—
157	Dic₁₀₁	C₂²×D₄	1	1	yes	characteristic
158	C₂×C₁₀₁	—	2	3	no	—
159	C₂×C₁₀₁	—	4	6	no	—
160	C₂×C₁₀₁	C₂³×D₄	1	1	yes	characteristic
161	D₁₀₁	—	2	1	no	—
162	D₁₀₁	—	2	3	no	—
163	D₁₀₁	—	4	2	no	—
164	#3584	—	101	1	no	maximal, Sylow
165	C₁₀₁	#3584	1	1	yes	characteristic, complemented, Sylow
166	#585	—	101	2	no	—
167	#578	—	101	3	no	—
168	#578	—	101	1	no	—
169	#577	—	101	2	no	—
170	#577	—	101	6	no	—
171	#576	—	101	3	no	—
172	#576	D₁₀₁	1	1	yes	characteristic, complemented
173	#575	—	101	6	no	—
174	C₂×#135	—	101	3	no	—
175	C₂×#135	—	101	3	no	—
176	C₂×#131	—	101	1	no	—
177	2₋¹⁺⁴	—	101	6	no	—
178	2₋¹⁺⁴	—	101	2	no	—
179	2₋¹⁺⁴	—	202	2	no	—
180	H(5,𝔽₂)	—	101	6	no	—
181	H(5,𝔽₂)	—	202	6	no	—
182	H(5,𝔽₂)	C₂×D₁₀₁	1	2	yes	complemented
183	C₂×#41	—	101	1	no	—
184	C₂×#41	—	101	6	no	—
185	C₂×#41	—	101	6	no	—
186	C₂×#41	—	202	6	no	—
187	C₂×#41	—	202	2	no	—
188	#137	—	101	6	no	—
189	#137	—	101	12	no	—
190	#137	—	101	6	no	—
191	AGL(1,ℤ/8)	—	101	6	no	—
192	AGL(1,ℤ/8)	—	101	12	no	—
193	AGL(1,ℤ/8)	C₂×D₁₀₁	1	6	yes	complemented
194	#135	—	101	6	no	—
195	#135	—	101	1	no	—
196	#135	—	101	6	no	—
197	#135	—	101	3	no	—
198	#135	—	101	6	no	—
199	#135	—	101	6	no	—
200	#135	—	101	6	no	—
201	#135	—	101	3	no	—
202	#135	C₂×D₁₀₁	1	3	yes	complemented
203	C₂×Q₁₆	—	101	3	no	—
204	C₂×Q₁₆	—	101	3	no	—
205	C₂×SD₁₆	—	101	6	no	—
206	C₂×SD₁₆	—	101	6	no	—
207	C₂×D₈	—	101	3	no	—
208	C₂×D₈	C₂×D₁₀₁	1	3	yes	complemented
209	#131	—	101	1	no	—
210	#131	—	101	3	no	—
211	#131	—	101	3	no	—
212	#131	C₂×D₁₀₁	1	1	yes	characteristic, complemented
213	C₂×M₁₆	—	101	3	no	—
214	C₂²×C₈	—	101	3	no	—
215	#41	—	2	2	no	—
216	#41	—	101	1	no	—
217	#41	—	101	3	no	—
218	#41	—	101	3	no	—
219	#41	—	101	2	no	—
220	#41	—	101	6	no	—
221	#41	—	101	12	no	—
222	#41	—	101	6	no	—
223	#41	—	202	6	no	—
224	#41	—	202	6	no	—
225	#41	—	202	6	no	—
226	#41	—	202	6	no	—
227	#41	—	202	6	no	—
228	#41	—	202	12	no	—
229	#41	—	202	6	no	—
230	#41	—	202	6	no	—
231	#41	C₂²×D₁₀₁	1	6	yes	—
232	#41	C₂²×D₁₀₁	1	1	yes	characteristic
233	C₂×Q₈	—	101	1	no	—
234	C₂×Q₈	—	101	6	no	—
235	C₂×Q₈	—	101	6	no	—
236	C₂×Q₈	—	202	6	no	—
237	C₂×Q₈	—	202	2	no	—
238	C₂×D₄	—	2	6	no	—
239	C₂×D₄	—	101	3	no	—
240	C₂×D₄	—	101	6	no	—
241	C₂×D₄	—	202	6	no	—
242	C₂×D₄	—	202	6	no	—
243	C₂×D₄	—	202	12	no	—
244	C₂×D₄	—	202	6	no	—
245	C₂×D₄	C₂²×D₁₀₁	1	6	yes	—
246	C₂²×C₄	—	101	3	no	—
247	C₂²×C₄	—	202	6	no	—
248	C₂²×C₄	—	202	6	no	—
249	Q₁₆	—	101	6	no	—
250	Q₁₆	—	101	3	no	—
251	Q₁₆	—	101	6	no	—
252	Q₁₆	C₂²×D₁₀₁	1	1	yes	characteristic
253	SD₁₆	—	101	2	no	—
254	SD₁₆	—	101	6	no	—
255	SD₁₆	—	101	12	no	—
256	SD₁₆	—	101	6	no	—
257	SD₁₆	C₂²×D₁₀₁	1	6	yes	—
258	D₈	—	101	6	no	—
259	D₈	—	101	1	no	—
260	D₈	C₂²×D₁₀₁	1	3	yes	—
261	D₈	C₂²×D₁₀₁	1	6	yes	—
262	M₁₆	—	101	3	no	—
263	M₁₆	—	101	6	no	—
264	M₁₆	C₂²×D₁₀₁	1	3	yes	—
265	C₂×C₈	—	101	3	no	—
266	C₂×C₈	—	101	1	no	—
267	C₂×C₈	—	101	3	no	—
268	C₂×C₈	—	101	3	no	—
269	C₂×C₈	—	101	3	no	—
270	C₂×C₈	C₂²×D₁₀₁	1	3	yes	—
271	C₂³	—	2	6	no	—
272	C₂³	—	101	3	no	—
273	C₂³	—	202	6	no	—
274	Q₈	—	101	3	no	—
275	Q₈	—	101	6	no	—
276	Q₈	—	202	6	no	—
277	Q₈	—	202	6	no	—
278	Q₈	D₄×D₁₀₁	1	1	yes	characteristic
279	Q₈	C₂³×D₁₀₁	1	2	yes	—
280	D₄	—	2	6	no	—
281	D₄	—	2	6	no	—
282	D₄	—	101	6	no	—
283	D₄	—	101	3	no	—
284	D₄	—	101	2	no	—
285	D₄	—	202	2	no	—
286	D₄	—	202	6	no	—
287	D₄	—	202	6	no	—
288	D₄	—	202	12	no	—
289	D₄	—	202	6	no	—
290	D₄	D₄×D₁₀₁	1	3	yes	—
291	D₄	C₂³×D₁₀₁	1	6	yes	—
292	C₂×C₄	—	2	6	no	—
293	C₂×C₄	—	101	3	no	—
294	C₂×C₄	—	101	3	no	—
295	C₂×C₄	—	101	3	no	—
296	C₂×C₄	—	101	1	no	—
297	C₂×C₄	—	101	3	no	—
298	C₂×C₄	—	202	2	no	—
299	C₂×C₄	—	202	6	no	—
300	C₂×C₄	—	202	6	no	—
301	C₂×C₄	—	202	12	no	—
302	C₂×C₄	—	202	6	no	—
303	C₂×C₄	—	202	6	no	—
304	C₂×C₄	C₂³×D₁₀₁	1	3	yes	—
305	C₈	—	101	1	no	—
306	C₈	—	101	3	no	—
307	C₈	C₂³×D₁₀₁	1	1	yes	characteristic
308	C₈	C₂³×D₁₀₁	1	3	yes	—
309	C₂²	—	2	6	no	—
310	C₂²	—	8	6	no	—
311	C₂²	—	808	6	no	—
312	C₂²	—	101	1	no	—
313	C₂²	—	101	3	no	—
314	C₂²	—	404	3	no	—
315	C₂²	—	202	2	no	—
316	C₂²	C₂×D₄×D₁₀₁	1	3	yes	—
317	C₄	—	2	2	no	—
318	C₄	—	101	1	no	—
319	C₄	—	101	3	no	—
320	C₄	—	202	6	no	—
321	C₄	C₂×D₄×D₁₀₁	1	3	yes	—
322	C₄	C₂⁴×D₁₀₁	1	1	yes	characteristic
323	C₂	—	2	3	no	—
324	C₂	—	4	6	no	—
325	C₂	—	404	2	no	—
326	C₂	—	202	3	no	—
327	C₂	—	202	1	no	—
328	C₂	C₂²×D₄×D₁₀₁	1	1	yes	characteristic

	subgroup	G/H
center	C₂	C₂²×D₄×D₁₀₁
commutator, cosocle	C₄×C₁₀₁	C₂⁵
Frattini	C₄	C₂⁴×D₁₀₁
Fitting	#576×C₁₀₁	C₂
socle	C₂×C₁₀₁	C₂³×D₄
hypercenter	#576	D₁₀₁
hypocenter	C₁₀₁	#3584
LCS	G ▷ C₄×C₁₀₁ ▷ C₂×C₁₀₁ ▷ C₁₀₁
UCS	C₁ ◁ C₂ ◁ #41 ◁ #576

Complements

Splittings G = N ⋊ H with N normal; H ≅ G/N.

N	H ≅ G/N	type
1. #20266012	C₂	semidirect
2. #20265994	C₂	semidirect
3. #20265991	C₂	semidirect
4. #20265988	C₂	semidirect
5. #20265987	C₂	semidirect
6. #20265978	C₂	semidirect
7. #20265973	C₂	semidirect
8. C₂×#15692274	C₂	semidirect
9. #135×D₁₀₁	C₂	semidirect
10. #131×D₁₀₁	C₂	semidirect
11. #576×C₁₀₁	C₂	semidirect
13. #15692302	C₂²	semidirect
14. #15692302	C₂²	semidirect
16. #15692300	C₂²	semidirect
18. C₂×#13108073	C₂²	semidirect
21. #41×D₁₀₁	C₂²	semidirect
22. #15692288	C₂²	semidirect
23. #15692285	C₂²	semidirect
24. #15692284	C₂²	semidirect
25. #15692281	C₂²	semidirect
26. #15692279	C₂²	semidirect
27. #15692278	C₂²	semidirect
28. #15692277	C₂²	semidirect
29. #15692276	C₂²	semidirect
30. #15692275	C₂²	semidirect
31. #15692274	C₂²	semidirect
32. #15692274	C₂²	semidirect
33. #15692274	C₂²	semidirect
34. #15692274	C₂²	semidirect
35. C₂×Dic₄₀₄	C₂²	semidirect
36. Q₁₆×D₁₀₁	C₂²	semidirect
37. C₂×#13108062	C₂²	semidirect
38. SD₁₆×D₁₀₁	C₂²	semidirect
39. D₈×D₁₀₁	C₂²	semidirect
40. #15692263	C₂²	semidirect
41. #15692262	C₂²	semidirect
42. #15692261	C₂²	semidirect
43. M₁₆×D₁₀₁	C₂²	semidirect
44. C₂×C₈×D₁₀₁	C₂²	semidirect
45. H(5,𝔽₂)×C₁₀₁	C₂²	semidirect
46. AGL(1,ℤ/8)×C₁₀₁	C₂²	semidirect
47. #135×C₁₀₁	C₂²	semidirect
48. C₂×D₈×C₁₀₁	C₂²	semidirect
49. #131×C₁₀₁	C₂²	semidirect
165. C₁₀₁	#3584	semidirect
172. #576	D₁₀₁	semidirect
182. H(5,𝔽₂)	C₂×D₁₀₁	semidirect
193. AGL(1,ℤ/8)	C₂×D₁₀₁	semidirect
202. #135	C₂×D₁₀₁	semidirect
208. C₂×D₈	C₂×D₁₀₁	semidirect
212. #131	C₂×D₁₀₁	semidirect

Maximal subgroups

By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.

subgroup	index
#20266012	2
#20265994	2
#20265991	2
#20265988	2
#20265987	2
#20265978	2
#20265973	2
C₂×#15692274	2
#135×D₁₀₁	2
#131×D₁₀₁	2
#576×C₁₀₁	2
#3584	101